987.410 Icosa Phase of Rationality

987.411 The 96 secondary gre at circles of the VE divide the chorded edge of the VE (which is the unit vector radius of synergetics) into rational linear fractions of the edge lengthi.e., \sfrac{1}{2}, \sfrac{3}{10}, \sfrac{1}{4}, \sfrac{1}{6}, \sfrac{1}{10} and these fractions embrace all the intercombinings of the first four prime numbers $1, 2, 3, and 5$.

987.412 For an illustration of how the four VE great circles of 60-degree central angles subdivide the central-angle chord increments, see Fig. 987.412.

987.413 Next recalling the jitterbug transformation of the VE into the icosa with its inherent incommensurability brought about by the 2 : \sqrt{2}= \sqrt{2}:1transformation ratio, and recognizing that the transformation was experimentally demonstrable by the constantly symmetrical contracting jitterbugging, we proceed to fractionate the icosahedron by the successive 15 great circles, six great circles (icosa type), and 10 great circles whose self-fractionation produces the S Modules* as well as the T and E Modules.

987.414 But it mu st be recalled that the experimentally demonstrable jitterbug model of transformation from VE to icosa can be accomplished through either a clockwise or counter clockwise twisting, which brings about 30 similar but positive and 30 negative omniintertriangulated vector edge results.

987.415 The midpoints of each of these two sets of 30 vertexes in turn provide the two alternate sets of 30 poles for the spin-halving of the 15 great circles of Symmetry #6, whose spinning in turn generates the 120 right spherical triangles (60 positive, 60 negative) of t he icosahedral system.

987.416 The 120 right triangles, evenly grouped into 30s pherical diamonds, are transformed into 30 planar diamonds of central angles iden tical to those of the 30 spherical diamonds of the 15 great circles of the icosa . When the radius to the center of the face of the rhombic triacontahedron equals .9994833324… of the unit vector radius of Synergetics (1.000), the rhombic triacontahedron has a tetravolume of 5 and each of its 120 T Quanta Modules h as a volume of one A Module. When the radius equals 1, the volume of the rhombic triacontahedron is slightly larger (5.007758029), and the corresponding E Module has a volume of 1.001551606 of the A Module. (See Sec. 986.540.) radius to radius = 2, or twice prime vector radius. This produces the stellated rhombic dodecahedron of tetravolume-12.

1. The stellated rhombic dodecahedron vertexes are congruent with the mid-edge points of the cube of tetravolume-24. A composite of three two-frequency Couplers (each individually of tetravolume-8) altogether comprises a star complex of tetravolume-12, sharing a common central rhombic dodeca domain of tetravolume-6. The stellated rhombic dodeca of tetravolume-12 is half the volume of the 24-tetravolume cube that inscribes it. (Compare the Duo-Tet Cube at Fig. 987.242A.)

See Sec. 988.